Armored Trooper. EVA Unit. Mobile Suit. Mortar Head. Jaeger. Heavy Gear. Ryude Knight.
They are Mecha.
While the term 'Mecha' was actually coined by the Japanese to reflect any kind of mechanized vehicle or device (a car is a mecha), it has become synonymous in Western culture with the large, humanoid, usually human piloted or otherwise operated robots seen in a vast number of Japanese Anime and Manga series.
For the most part, mecha, being machines of some sort, are largely relegated to Science Fiction settings, although a more than casual glance across the Anime/Manga spectrum reveals mecha in such diverse genres as medieval fantasy and steampunk to World War II alternate history and modern day 'superheroes'.
The granddaddy of all Mecha RPGs is, well, sadly it's Battletech. Unfortunately the giant robots of Battletech have little to do with Japanese entertainment fiction mecha. In Battletech, the very existence of the Battlemechs they use seem questionable at best. They overheat more often then tanks in the setting, are slower than all the aircraft, and don't generally seem as versatile as you'd imagine a robot with legs and arms would be. In attempt to capture the 'realism' of what giant, humanoid robots would be like (wait...I need to stop laughing. Realism. Giant Robots. Hold on, I can't breathe. OK. Whew.), they took away the beauty of having giant robots in your setting in the first place, IMHO.
The first game to truly 'get it', and the only name you'll ever really need to know for giant robot gaming goodness, was (and is) Mekton.
The creation of Mike "Right on Time, Ahead of His Time" Pondsmith and produced by R. Talsorian Games, Mekton wasn't inspired by the art of giant robots from Japan, it was ABOUT the giant robots from Japan. At the same time, it was very much about the people who piloted said robots, especially in its second and third incarnations (Mekton II and Mekton Zeta respectively).
Nailing the style and feel of Japanese animation and comics perfectly, as well as hitting that oh-so-hard-to-hit sweet spot between simplicity and crunch*, Mekton became a major favorite among Players and Gamemasters alike in the gaming circles in which I roamed. The flexibility and versitility of the rules system meant that it was adaptable to many different campaign concepts.
While the default for many is the military style giant robot war story seen in Votoms, Macross and the various Gundam series, the game also makes the superheroic mecha of Giant Robo, The Big O, Mazinger Z and others possible. With a few of the supplements (but in all honesty just a little imagination and a minimum of effort) you can easily create powered armor, cars, starfighters, massive space battlecruisers and even fortresses and space stations.
Practical? No. Easy to construct? Pretty much, yeah.
A number of other mech games have come out since the original 1984 release of Mekton and, more importantly perhaps, the 1994 release of Mekton Z. The vast majority of these have been very much flashes-in-the-pan. The most popular and successful non-Mekton mecha game that really got the feel down right was Heavy Gear and the Jovian Chronicles, produced by Canadian based Dream Pod 9 and utilizing their Silhouette System. Heavy Gear was the most commerically successful, also generating a huge line of miniatures and a CGI animated cartoon.
Other notable mecha games include the mecha supplement book for GURPS (GURPS Mecha if I am not mistaken), Bliss Stage (a very cool and clever indie game patterned after Neon Genesis Evangelion) and to a fairly significant extent RIFTS (which is full of mecha, from battlesuits to gigantic striding metal menaces).
I could easily go on forever discussing this element of Japanese Amime/Manga and to that end I will follow this up with a post detailing some of the giant robot campaigns I have run in the past. For a long time, to many people who were only generally familiar with Japanese pop culture entertainment, all Anime and Manga was Mecha Anime and Manga. While the field is obviously far more diverse than that, it remains a favorite sub-genre of yours truly, possibly due to my general fascination with robots outside of gaming.
If you're interested, here are a few older posts on the subject of Mecha and Mekton:
Watashi-tachi wa tanoshinde imasu ka?